Book ChapterDOI
Stochastic Differential Equations
Ioannis Karatzas,Steven E. Shreve +1 more
- pp 281-398
TLDR
In this paper, the authors explore questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties, using diffusion processes as a model of a Markov process with continuous sample paths.Abstract:
We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.read more
Citations
More filters
Journal ArticleDOI
A Brief History of Generative Models for Power Law and Lognormal Distributions
TL;DR: A rich and long history is found of how lognormal distributions have arisen as a possible alternative to power law distributions across many fields, focusing on underlying generative models that lead to these distributions.
Journal ArticleDOI
High-Order Collocation Methods for Differential Equations with Random Inputs
Dongbin Xiu,Jan S. Hesthaven +1 more
TL;DR: A high-order stochastic collocation approach is proposed, which takes advantage of an assumption of smoothness of the solution in random space to achieve fast convergence and requires only repetitive runs of an existing deterministic solver, similar to Monte Carlo methods.
Book
Uncertainty Theory
TL;DR: Mathematicians, researchers, engineers, designers, and students in the field of mathematics, information science, operations research, industrial engineering, computer science, artificial intelligence, and management science will find this work a stimulating and useful reference.
Posted Content
Score-Based Generative Modeling through Stochastic Differential Equations
TL;DR: This work presents a stochastic differential equation (SDE) that smoothly transforms a complex data distribution to a known prior distribution by slowly injecting noise, and a corresponding reverse-time SDE that transforms the prior distribution back into the data distribution by Slowly removing the noise.
Journal ArticleDOI
Hidden Markov processes
Yariv Ephraim,Neri Merhav +1 more
TL;DR: An overview of statistical and information-theoretic aspects of hidden Markov processes (HMPs) is presented and consistency and asymptotic normality of the maximum-likelihood parameter estimator were proved under some mild conditions.
References
More filters
Book
Introduction to Econophysics: Correlations and Complexity in Finance
TL;DR: Economists and workers in the financial world will find useful the presentation of empirical analysis methods and well-formulated theoretical tools that might help describe systems composed of a huge number of interacting subsystems.
Journal ArticleDOI
A Brief History of Generative Models for Power Law and Lognormal Distributions
TL;DR: A rich and long history is found of how lognormal distributions have arisen as a possible alternative to power law distributions across many fields, focusing on underlying generative models that lead to these distributions.
Journal ArticleDOI
High-Order Collocation Methods for Differential Equations with Random Inputs
Dongbin Xiu,Jan S. Hesthaven +1 more
TL;DR: A high-order stochastic collocation approach is proposed, which takes advantage of an assumption of smoothness of the solution in random space to achieve fast convergence and requires only repetitive runs of an existing deterministic solver, similar to Monte Carlo methods.
Book
Uncertainty Theory
TL;DR: Mathematicians, researchers, engineers, designers, and students in the field of mathematics, information science, operations research, industrial engineering, computer science, artificial intelligence, and management science will find this work a stimulating and useful reference.
Posted Content
Score-Based Generative Modeling through Stochastic Differential Equations
TL;DR: This work presents a stochastic differential equation (SDE) that smoothly transforms a complex data distribution to a known prior distribution by slowly injecting noise, and a corresponding reverse-time SDE that transforms the prior distribution back into the data distribution by Slowly removing the noise.